New supercharacter theory for Sylow subgroups in orthogonal and symplectic groups
aa r X i v : . [ m a t h . R T ] A ug New supercharacter theory for Sylow subgroups in orthogonal andsymplectic groups
A.N.Panov ∗ § The notion of a supercharacter theory was suggested by P.Diaconis andI.M.Isaaks in the paper [1]. A priori every group affords several superchar-acter theories. One of examples of a supercharacter theroy is the theoryof irreducible characters. Since for some groups (such as the unitriangu-lar group, the Sylow subgroups in symplectic and orthogonal groups, theparabolic subgroups and others) the problem of classification of irreduciblecharacters (representations) remains to be a very complicated ”wild” prob-lem, it appears reasonable to replace this problem by the problem of con-struction of a supercharacter theory, which provides the best approximationof theory of irreducible characters.Let us formulate the definition of a supercharacter theory following thepaper [1]. Let G be a finite group, 1 ∈ G be a unit element. Let Ch = { χ , . . . , χ N } be a system of complex characters (representations) of thegroup G . Definition 1 . The system of characters Ch determines a supercharactertheory of G if there exists a partition K = { K , . . . , K N } of the group G satisfying the following conditions:S1) the characters of Ch are pairwise disjoint (orthogonal);S2) each character χ i are constant on each subset K j ;S3) { } ∈ K .Under this definition each character of Ch is referred to as a supercharacter ,each subset of K a superclass . Observe that the number of supercharactersis equal to the number of superclasses. The square table { χ i ( K j ) } is calleda supercharacter table. For each supercharacter χ i , consider its support X i (the subset of all irre-ducible constituents of χ i ). Observe that the condition S3) of Definition 1 ∗ The classification of superclasses (section 2) is supported by the RFBR grant 16-01-00154-a; the construction of super-character theory (section 3) is supported by grant RSF-DFG grant 16-41-1013 ay be replaced by following condition:S3’) The system of subsets X = { X , . . . , X N } is a partition of the system ofirreducible characters Irr( G ). Moreover, here each character χ i differs fromthe character σ i = P ψ ∈ X i ψ (1) ψ by a constant factor (see [1, 9, 10]).For the unitriangular group UT( m, F q ), the suitable supercharacter theorywas constructed in the series of papers of C.A.M. Andr´e [2, 3, 4]. This theorywas generated for the algebra groups by P.Diaconis and I.M.Isaaks [1]. Bydefinition, an algebra group is a group of the form G = 1 + J , where J is anassociative finite dimensional nilpotent algebra. Superclasses in the algebragroup G are the equivalence classes for the equivalence relation: g ∼ g ′ ,where g = 1 + x and g ′ = 1 + x ′ , if there exist a, b ∈ G such that x ′ = axb .The similar relation is defined for J ∗ : by definition, λ ∼ λ ′ if there exist a, b ∈ G such that λ ′ = aλb (here aλb ( x ) = λ ( bxa )).Fix a nontrivial character t → ε t of the additive group of the field F q withvalues in the group of invertible elements of the field C . Supercharacters χ λ of a given algebra group are the induced characters from linear characters ξ λ (1 + x ) = ε λ ( x ) of right stabilizers of λ ∈ J ∗ . The sets of characters { χ λ } and classes { K ( g ) } ,where λ and g run through the set of representatives of equivalence classesof J ∗ and G respectively, give rise to a supercharacter theory of the algebragroup G . Supercharacters χ λ afford the analog of A.A.Krillov formula (see[1, 10]): χ λ (1 + x ) = | Gλ || GλG | X µ ∈ GλG ε µ ( x ) . An unipotent group is not an algebra group in general. The outlinedmethod is not valid for unipotent groups. In this paper, we propose thenew approach which can be applied for a large class of unipotent groups,hypothetically. The application of this approach for the Sylow subgroups oforthogonal and symplectic groups enables to construct the supercharactertheory (see Theorem 19), which is a bit better then the one suggested inthe papers [5, 6, 7, 8].Let us present the content of this approach. Let U be an unipotent groupthat is a semidirect product U = U U with the normal subgroup U . Sup-pose that U is an algebra group, i.e. U = 1 + u , where u is an associatedfinite dimensional nilpotent algebra. The Lie algebra u of the group U is adirect sum of two subalgebras u = u ⊕ u , where u is an associated algebra,and u = Lie( U ) is an ideal in u . Since U is an algebra group, for any a ∈ U and x ∈ u , the elements ℓ a ( x ) = ax and r a ( x ) = x a also belongs o u . The left and right actions of U on u can be extended to the actionson u as follows ℓ a ( x ) = ℓ a ( x ) + Ad a ( x ) ,r a ( x ) = r a ( x ) + x , (1)where a ∈ U and x = x + x , x ∈ u , x ∈ U . Observe that the leftand right actions of the subgroup U commute, and ℓ a r − a ( x ) = Ad a ( x ) , (2)where Ad a is the adjoint operator for a ∈ U . Definition 2 . Let x, x ′ ∈ u . The element x is equivalent to x ′ if there existsa chain of transformations of forms1) x → ℓ a ( x ), where a ∈ U ,2) x → Ad u ( x ), where u ∈ U ,that maps x to x ′ .Because of (2) we may substitute r a for ℓ a in 1).Fix Ad-invariant bijective map f : U → u , f (1) = 0. As a map f wecan take the logarithm ln (it requires strong restrictions of characteristic ofthe field; see below Definition 12 for Sylow subgroups in orthogonal andsymplectic groups). Introduce the equivalence relation on U as follows. Definition 3 . Two elements u and u of the group U are equivalent if theelements f ( u ) and f ( u ) from u are equivalent in the sense of Definition 2.Consider the equivalence classes { K ( u ) } ; hypothetically they are super-classes for some supercharacter theory.Let us define the left and right actions of U on the dual space u ∗ by theformulas ℓ ∗ a λ ( x ) = λ ( r a ( x )) ,r ∗ a λ ( x ) = λ ( ℓ a ( x )) . The equivalence relation on u ∗ is defined similarly to 2 for u . Definition 4 . Let λ, λ ′ ∈ u ∗ . The element λ is equivalent to λ ′ if there existsa chain of transformations of forms1) λ → ℓ ∗ a ( λ ), where a ∈ U ,2) λ → Ad ∗ u ( λ ), where u ∈ U ,that maps x to x ′ .As above ℓ ∗ a ( r ∗ a ) − λ = Ad ∗ a λ ; in definition, we may substitute r ∗ a for ℓ ∗ a . Denoteby O ( λ ) the equivalence class of λ ∈ u ∗ .In this paper, we present the classification of equivalence classes in u , U and u ∗ for the Sylow subgroups in orthogonal and symplectic groups (seeTheorems 10, 11, 13). Conjecture 5 . There exists a system of characters of a given finite unipotent roup U of the form χ λ ( u ) = c ( λ ) X µ ∈O ( λ ) ε µ ( f ( u )) , where c ( λ ) ∈ C , c ( λ ) = 0 , (3)such that along with the partition of the group U into the classes { K ( u ) } ,where λ and u run through the sets of representatives of equivalence classesin u ∗ and U respectively, give rise to a supercharacter theory of the group U . Remark 6 (see [8]). Observe that the formula (3) defines the system oforthogonal functions on U (since the characters { ε λ ( x ) } of the abelian group u are pairwise orthogonal). Easy to see that the functions (3) are constanton the classes K ( u ). From f (1) = 0 it follows K (1) = 1. So, the functions(3) always fulfil the conditions S1, S2, S3. The main problem is to proveexistence of constants c ( λ ) such that the formula (3) defines a character ofsome representation of the group U . § The unitriangular group G = UT( m, F q ) consists of all upper triangularmatrices of order m with ones on the diagonal and entries from the finitefield F q . Assume that the characteristic of field p >
2. The Lie algebra ofthe unitriangular group g = ut ( m, F q ) consists of upper triangular matriceswith zeros on the diagonal.Consider the matrices I n = . . . ... . . . ... . . . and J n = (cid:18) I n − I n (cid:19) . Let m denote the dimension of standard representation of Lie algebras oftypes B n , C n , and D n . That is m = 2 n + 1 for B n , and m = 2 n for C n and D n . The matrix algebra Mat( m, F q ) affords the involutive antiautomorphism X → X † , where X † = I m X t I m for B n and D n , and X † = J n X t J n for C n .The standard Sylow subgroup U in orthogonal and symplectic group con-sists of all g ∈ G obeying g † = g − . Respectively, its Lie algebra u = { x ∈ g : x † = − x } .We denote by X τ the matrix transposed to X with respect to the seconddiagonal. The Lie algebra u for C n and D n consists of matrices of the form u = (cid:26)(cid:18) X X − X τ (cid:19)(cid:27) , (4) here X ∈ ut ( n, F q ), X τ = X for C n and X τ = − X for D n . The Liealgebra u is a sum of two subalgebras u = u + u , where u = (cid:26)(cid:18) X − X τ (cid:19)(cid:27) , u = (cid:26)(cid:18) X (cid:19)(cid:27) . The subalgebra u is an ideal in u , and u is isomorphic to ut ( n, F q ), and,therefore, it has a natural structure of associative algebra.The group U is a semidirect product U = U U , where U = (cid:26)(cid:18) E B E (cid:19)(cid:27) and U = (cid:26)(cid:18) A
00 ( A τ ) − (cid:19)(cid:27) , (5) B τ = B for C n and B τ = − B for D n , and A ∈ UT( n, F q ). The subgroup U is isomorphic to UT( n, F q ), and, therefore, it is an algebra group.In the case B n , the Lie algebra u consists of matrices of the form X X X − X τ − X τ (6)where X ∈ ut ( n, F q ), X is a n × X is a n × n matrixand X τ = − X . As above the Lie algebra u is a sum of two subalgebras u = u + u , where u = X − X τ , u = X X − X τ . The subalgebra u is an ideal in u , and u is isomorphic to ut ( n, F q ), and,therefore, it has a natural structure of associative algebra.The group U decomposes U = U U , where U = E v − vv τ + B − v τ E and U = A A τ ) − , (7) B τ = − B , and v is a n -column. The subgroup U is isomorphic to UT( n, F q ),and it is an algebra group.Let us define the left and right actions of the subgroup U on u followingthe formula (1). For C n and D n , a = diag( A, ( A τ ) − ) and x ∈ u , we have ℓ a ( x ) = ℓ a ( x ) = ℓ a ( x ) + Ad a ( x ) = (cid:18) AX AX A τ − X A τ (cid:19) r a ( x ) = r a ( x ) + x = (cid:18) X A X − A τ X τ (cid:19) . bserve ℓ a ( x ) = (cid:18) A E (cid:19) (cid:18) X X − X τ (cid:19) (cid:18) E A τ (cid:19) = a xa † , where a = diag( A, E ); r a ( x ) = (cid:18) E A τ (cid:19) (cid:18) X X − X τ (cid:19) (cid:18) A E (cid:19) = a xa † , where a = diag( E, A τ ) . For B n , a = diag( A, , ( A τ ) − ), and x ∈ u , we have ℓ a ( x ) = ℓ a ( x ) + Ad a ( x ) = AX AX AX A τ − X τ A τ − X τ A τ ,r a ( x ) = r a ( x ) = r a ( x ) + x = X A X X − X τ − A τ X τ . Observe ℓ a ( x ) = A E X X X − X τ − X τ E A τ = a xa † , where a = diag( A, , E ); r a ( x ) = E A τ X X X − X τ − X τ A E = a xa † , where a = diag( E, , A τ ) . Denote by G ◦ the subgroup in G = UT( n, F q ) generated by the sub-group U and the matrices diag( A , A ) in the case C n and D n (respectively,diag( A , , A ) in the case B n ). Remark . The elements x, x ′ ∈ u are equivalent if and only if there exists g ∈ G ◦ such that x ′ = gxg † .In referred to above papers [5, 6, 7, 8], the equivalence relation is a bitcoarser: x ∼ x ′ if there exists g ∈ G such that x ′ = gxg † .Denote by H ◦ the subgroup in G = UT( n, F q ) generated by the sub-group U and the matrices diag( A , E ) in the case C n and D n (respectively,diag( A , , E ) in the case B n ).Let us describe the subgroups G ◦ and H ◦ . Proposition 7 . 1) In the case B n , G ◦ = A A A A A A and H ◦ = A A A A A ′ E , (8) here A , A ∈ UT( n − , F q ), A , A , A are an arbitrary matrices of sizes( n − ×
3, ( n − × ( n − × ( n −
1) respectively, A ′ is an arbitrary3 × n matrix with zero last row, E is the unit n × n matrix, A is the 3 × c − c − c , c ∈ F q . (9)2) In the case C n , the subgroup G ◦ coincides with G and H ◦ = (cid:26)(cid:18) A A E (cid:19)(cid:27) .
3) In the case D n , G ◦ = (cid:26)(cid:18) A A A (cid:19)(cid:27) and H ◦ = (cid:26)(cid:18) A A E (cid:19)(cid:27) , (10)where A , A ∈ UT( n, F q ), and A is an arbitrary n × n matrix of the form ∗ ∗ · · · ∗ ... ... . . . ...c ∗ · · · ∗ − c · · · ∗ , c ∈ F q . (11) Proof . We shall prove for the subgroup H ◦ (for G ◦ similarly). Case C n . The subgroup H ◦ is generated by the matrices of the form (cid:18) A E (cid:19) , where A ∈ UT( n, F q ) and (cid:18) E B E (cid:19) with B τ = B . Then thematrix (cid:18) A E (cid:19) (cid:18) E B E (cid:19) (cid:18) A − E (cid:19) = (cid:18) E AB E (cid:19) also belongs to H ◦ . Easy to verify that the linear subspace, spanned by thematrices of the form AB , where A ∈ UT( n, F q ) and B τ = B , coincides withMat( n, F q ). This proves statement 2). Case D n . It is treated similarly. Easy to verify that the linear subspace,spanned by the matrices of the form AB , where A ∈ UT( n, F q ) and B τ = − B , coincides with the subspace of matrices of the form (11). This provesstatement 3). Case B n . For any two n -columns v and v we consider the matrix M ( v , v ) = E v − v v τ − v τ E . (12)The group H ◦ is generated by the matrices diag( A, , E ) with A ∈ UT( n, F q ), he matrices M ( v, v ), where v is a n -column, and F ( B ) = B , where B τ = − B . Analogically the case D n one can show that the matricesof the form F ( B ), where B is a matrix (11), belong to H ◦ .The equality A E · M ( v, v ) · A − E = M ( Av, v )implies the subgroup H ◦ contains all matrices of the form M ( v , v ) for allcolumns v = β ...β n , v = β ′ ...β ′ n such that β n = β ′ n = 0 . For any n -columns v , v , d , d we have an equality M ( v , v ) M ( d , d ) = M ( v + d , v + d ) F ( B ) , (13)where B = ( − v d τ + d v τ ).The equality (13) implies the subgroup H ◦ contains all matrices of theform F ( B ), where B = ( b ij ) is an arbitrary n × n matrix with b n = 0.Finally, applying (13) one can verify H ◦ contains all matrices of the form M ( v , v ), where β n = β ′ n . This follows statement 1). ✷ Let us describe the equivalence classes in u and u ∗ (see Definitions 2 and4). Order the set of integers of the segment [ − n, n ] as follows1 ≺ . . . ≺ n ≺ ≺ − n ≺ . . . ≺ − . Denote by ∆ + the set of following integer pairs from [ − n, n ]:for B n ∆ + = { ( i, j ) : 1 i n, i ≺ j ≺ − i } , for C n ∆ + = { ( i, j ) : 1 i n, i ≺ j − i, j = 0 } , for D n ∆ + = { ( i, j ) : 1 i n, i ≺ j ≺ − i, j = 0 } . We refer to elements from ∆ + as positive roots , and to ∆ + as the set ofpositive roots . or any positive root α = ( i, j ) ∈ ∆ + , we call i a row number (denote i = row( α )) and j a column number (denote j = col( α )). Definition 8 . The subset
D ⊂ ∆ + is called basic if there is no more thanone root from D in each row and each column.The other name is the set of rook placement type. Definition 9 . We refer to a subset
D ⊂ ∆ + as quasibasic if 1) there is nomore than one root of D in any column;2) there is no more than one root of D in any row except the cases of B n , D n and pairs of roots ( i, n ) and ( i, − n ).So, any quasibasic subset in C n is a basic subset. For any positive root α = ( i, j ) denote α ′ = ( − j, − i ) (according to definition α ′ is not a positiveroot). For any matrix x = ( x α ) ∈ u the entices x α and x α ′ differs by a sign x α = ǫ ( α ) x α ′ .The Lie algebra g has the standard basis { E ij : 1 i < j m } . The Liealgebra u also has the standard basis {E α = E α + ǫ ( α ) E α ′ } . By a pair ( D , φ ),where D is a quasibasic subset of ∆ + and a map φ : D → F ∗ q , we define theelement x D ,φ = X α ∈D φ ( α ) E α . Theorem 10 . 1) Each element x ∈ u is equivalent to some element x D ,φ . 2)The pair ( D , φ ) is uniquely determined by x . Proof . 1) Applying transformations x → gxg † , g ∈ G ◦ , we are able toobtain more zeros in the matrix x , and finally get x D ,φ .2) Let us show that ( D , φ ) is uniquely determined by x . Suppose that theelements x D ,φ and x D ′ ,φ ′ are equivalent.For any positive root ( i, j ), we consider the submatrix Mat ij ( x ) of x withsystems of rows and columns { k : i k j } . Easy to see that if x ∼ x ′ ,then the submatrices Mat ij ( x ) and Mat ij ( x ′ ) have equal ranks.Suppose that D and D ′ are basic subsets. The equality of ranks impliesthat D = D ′ . For each root α = ( i, j ) ∈ D , we consider the subset D α ⊂ D that consists of ( k, m ) ∈ D , where k < i and m j . For α ∈ D we considerthe minor M α of the matrix x with systems of rows row( D α ) and columnscol( D α ). It is not difficult to show that the equivalence of matrices x D ,φ and x D ,φ ′ implies M α ( x D ,φ ) = M α ( x D ,φ ′ ). Hence φ = φ ′ .Assume that the quasibasic subset D is not basic. In this case, u is of thetype B n or D n , and D contains the pair of roots β = ( i, n ) and β = ( i, − n ).For D n the statement can be proves similarly the case of basic subset.Consider the case of B n . The equality of ranks of matrices Mat ij impliesthat D \ { β } = D ′ \ { β } . For α ∈ D \ { β } , we consider the system of roots D α ∈ D that consists of ( k, m ) ∈ D \ { β } , where k < i and m j . By the ubset D α we define the minor M α as above. Since M α ( x D ,φ ) = M α ( x D ,φ ′ ), itfollows φ ( α ) = φ ′ ( α ) for each α ∈ D \ { β } .It remains to show β ∈ D ′ and φ ( β ) = φ ′ ( β ). For β , we construct theroot system e D β that coincides with D β if there is no root ( k, i < k n ,in D ; if such root exists, then e D β = D β ∪ ( k, f M β .For β = ( i, e D β that coincides with( D β \ { β } ) ∪ { β } if there is no root ( k, i < k n , in D ; if such root exists, then e D β = D β ∪ { ( k, − n ) } . As above we define the minor f M β .Take e D β = D β \ { β } and consider the corresponding minor f M β . Onecan show that polynomial I = f M β f M β + 12 f M β is constant on the equivalence class of element x = x D ,φ . As x ∼ x ′ , where x ′ = x D ′ ,φ ′ , we have I ( x ) = I ( x ′ ). Observe that f M β ( x ) = f M β ( x ′ ) = 0, and f M β ( x ) = f M β ( x ′ ) = 0. Therefore, f M β ( x ) = f M β ( x ′ ) = 0. Hence β ∈ D ′ ,the values of φ and φ ′ on the root β coincide. ✷ The dual space g ∗ has the dual basis { E ∗ ij : 1 i < j m } . The dualspace u ∗ has the basis { E ∗ α + ǫ ( α ) E ∗ α ′ } . For ( D , φ ), we construct the element λ D ,φ = X α ∈D φ ( α ) ( E ∗ α + ǫ ( α ) E ∗ α ′ ) . Theorem 11 . 1) Each element λ ∈ u is equivalent to some element λ D ,φ . 2)The pair ( D , φ ) is uniquely determined by λ . Proof . The proof is similar to the one of Theorem 10. ✷ Turn to the equivalence relation on the group U . We take the Springermap as a map f in Definition 3. Definition 12 . A map f : G → g is called Springer map if f is a bijectionand it obeys the following conditions:1) f ( U ) = u ,2) there exist a , a , . . . from the field F q such that f (1 + x ) = x + a x + a x + . . . for any x ∈ g .Examples of Springer map are as follows:1) the logarithm map ln(1 + x ) = P ∞ i =1 ( − i +1 x i i (it requires strong restric-tions on the characteristic of field);2) Cayley’s map f (1 + x ) = xx +2 (for char F q = 2). enote u D ,φ = f − ( x D ,φ ). Classification of equivalence classes in group U follows from Theorem 10. Theorem 13 . 1) Each element u ∈ U is equivalent to some u D ,φ . 2) Thepair ( D , φ ) is uniquely determined by u . § In the case C n and D n , the group G ◦ is an algebra group, and constructionof supercharacters for the group U doesn’t differ from the approach of paper[8].In the case B n , the group G ◦ is not an algebra group. Consider the sub-group S of all matrices diag( E, M, E ), where M is a matrix of the type (9).The group G ◦ is a semidirect product G ◦ = SG ⋄ , where G ⋄ is a subgroup of G ◦ that consists of matrices of the form (8) with A = E . The subgroup G ⋄ is an algebra group G ⋄ = 1 + g ⋄ , where g ⋄ is the Lie subalgebra of the group G ⋄ .The group U is also a semidirect product U = SU ⋄ , where U ⋄ = U ∩ G ⋄ .The Lie algebra u decomposes u = s ⊕ u ⋄ , where s is the one dimensional Liealgebra spanned by the vector E n, .Take H ⋄ = G ⋄ ∩ H ◦ . The subgroup H ⋄ is an algebra group and H ◦ = SH ⋄ .Easy to see that in each case B n , C n , D n we have G ◦ = U H ⋄ (see [8, Lemma6.6]).For any linear form η ∈ ( g ⋄ ) ∗ , we define the following associative subalge-bras in g ⋄ :1) r ⋄ η = { x ∈ g ⋄ : η ( xy ) = 0 for any y ∈ h ⋄ } ;2) ℓ ⋄ η = { x ∈ g ⋄ : η ( y † x ) = 0 for any y ∈ h ⋄ } ;3) g ⋄ η = r ⋄ η ∩ ℓ ⋄ η .Since ( r ⋄ η ) † = ℓ ⋄ η , we have ( g ⋄ η ) † = g ⋄ η . The subgroup G ◦ contains thealgebra subgroups R ⋄ η = 1 + r ⋄ η , L ⋄ η = 1 + ℓ ⋄ η and G ⋄ η = 1 + g ⋄ η . Lemma 14 [8, Lemma 6.3]. η ( xy ) = 0 for any x, y ∈ g ⋄ η . Proof . Present x in the form x = x ′ + x ′′ , where x ′ ij = 0 for 1 j
0, and x ′′ ij = 0 for 0 ≺ j −
1. Present y in the form y = y ′ + y ′′ , where y ′ ij = 0 for0 ≺ i −
1, and y ′′ ij = 0 for 1 i
0. Then x ′ ∈ ( h ⋄ ) † , y ′ ∈ h ⋄ , x ′ y ′ = 0, x ′′ y ′′ = 0. The equality xy = x ′ y + xy ′ implies η ( xy ) = 0. ✷ Let λ ∈ ( u ⋄ ) ∗ and η ∈ g ⋄ be such that η † = − η and the restriction of η on u ⋄ coincides with λ . Define u ⋄ λ = u ∩ g ⋄ η and U ⋄ λ = U ∩ G ⋄ η . For x ∈ u ⋄ , we have η ( y † x ) = − η † ( y † x ) = − η (( y † x ) † ) = − η ( x † y ) = η ( xy ) . (14) t follows u ∩ r ⋄ η = u ∩ ℓ ⋄ η = u ⋄ λ .By Lemma 14, the restriction of λ on the Lie algebra u ⋄ λ is its characterwith values in the field F q .Let π ⋄ be the natural projection ( u ⋄ ) ∗ → ( u ⋄ λ ) ∗ . The following statementfollows from the paper [8]. For readers convenience we present it with com-plete proof. Lemma 15 . For any λ ∈ ( u ⋄ ) ∗ the fiber ( π ⋄ ) − π ⋄ ( λ ) equals to H ⋄ (cid:5) λ . Proof . Item 1 . Let η ∈ ( g ⋄ ) ∗ and P is the natural projection ( g ⋄ ) ∗ → ( r ⋄ η ) ∗ .Let us show that P − P ( η ) = H ⋄ η (here hη ( x ) = η ( xh ) is the left action h ∈ H ⋄ on η ). Indeed, P − P ( η ) = η + ( r ⋄ η ) ⊥ . The definition of r ⋄ η implies r ⋄ η = { x ∈ g ⋄ : yη ( x ) = 0 for any y ∈ h ⋄ } = ( h ⋄ η ) ⊥ . Hence ( r ⋄ η ) ⊥ = h ⋄ η and P − P ( η ) = η + h ⋄ η = (1 + h ⋄ ) η = H ⋄ η . Item 2 . Let Π be the natural projection ( g ⋄ ) ∗ → ( u ⋄ ) ∗ . Let η † = − η and λ = Π( η ). Let us show that Π( H ⋄ η ) = H ⋄ (cid:5) λ .Really, for h = 1 + y ∈ H ⋄ and x ∈ u ⋄ , we obtain h (cid:5) λ ( x ) = ( hλh † )( x ) = λ ( h † xh ) = λ ((1 + y ) † x (1 + y )) = λ ( x ) + λ ( y † x + xy ) + λ ( y † xy ) . Observe that y † xy = 0 for any x ∈ g and y ∈ h ⋄ . Applying the equality(14) we obtain h (cid:5) λ ( x ) = η ( x )+2 η ( xy ) = (1+2 y ) η ( x ) . Hence H ⋄ (cid:5) λ = Π( H ⋄ η );this proves statement 2. Item 3 . Since u ⋄ λ = u ⋄ ∩ r ⋄ λ , we have( π ⋄ ) − π ⋄ ( λ ) = Π( P − P ( η )) = Π( H ⋄ η ) = H ⋄ (cid:5) λ. ✷ Let λ = λ D ,φ and η = η D ,φ is the element ( g ) ∗ such that Π( η ) = λ and η † = − η . Lemma 16 . Let λ and η are defined by D , φ as above. Then sg ⋄ λ ⊂ g ⋄ λ and g ⋄ λ s ⊂ g ⋄ λ . Proof . Let η ∈ ( g ⋄ ) ∗ and Π( η ) = λ . For each α = ( i, j ) ∈ ∆ + , where1 i < n and i ≺ j , we define the subalgebra r ⋄ α that consists of matrices x ∈ g ⋄ obeying the conditions:1) if 1 j < n , then x ik = 0 for all i < k < j ;2) if j ∈ { n, , − n } , then x ik = 0 for all i < k < n ;3) if − n < j −
1, then x ik = 0 for all i ≺ k ≺ − n , and x − j,k = 0 for all − j ≺ k ≺ − n .Easy to see that the subalgebra r ⋄ α is invariant with respect to the leftand right multiplication by s . The subalgebra r ⋄ η coincides with intersectionof subalgebras r ⋄ α over all α ∈ D . The subalgebra r ⋄ η is also invariant withrespect to the left and right multiplication by s . ✷ Define the subalgebra u λ in each of the following cases separately.1) D doesn’t contain any roots of the form ( i,
0) and ( i, n ). In this case, e take g λ = s ⊕ g ⋄ λ and G λ = SG ⋄ λ . Denote u λ = g λ ∩ u = s ⊕ u ⋄ λ . Take U λ = G λ ∩ U = SU ⋄ λ .2) D contains ( i,
0) or ( i, n ). In this case, we define g λ = g ⋄ λ , u λ = u ⋄ λ , and G λ = U ⋄ λ .The associative subalgebra h ⋄ is an ideal in the associative algebra g .Therefore, h λ = g λ + h ⋄ is its associative subalgebra in g . Then the subgroup H λ = G λ H = 1 + h λ is an algebra group in G .Let π be the natural projection u ∗ → u ∗ λ . Lemma 17 . 1) For any λ = λ D ,φ , the fiber π − π ( λ ) coincides with H λ (cid:5) λ .2) The formula ξ λ ( u ) = ε λ ( f ( u )) (15)defines a character of the subgroup U λ . Proof . Item 1. D doesn’t contain any roots of the form ( i,
0) and ( i, n ).Then η ( s x ) = η ( x s ) = 0 for any x ∈ g . From Lemma 14 it follows that ξ λ ( g ) = ε λ ( f ( g )) is a character of G λ ; this proves statement 2).By direct calculations, we obtain hηh † ( x ) = η ( x ) for any x ∈ u λ and h ∈ H λ . The Lemma 15 implies statement 1). Item 2. D contains one of roots of type ( i,
0) or ( i, n ). Denote this rootfrom D by γ . If γ = ( i, z = E − n, − i ; if γ = ( i, − n ), then wetake z = E , − i . In each case z ∈ g λ ⊂ h λ and η ( E n z ) = ± φ ( γ ) = 0. Theelement h t = 1 + tz belongs to H λ for any t ∈ F q . By direct calculations, weobtain h t ηh † t ( x ) = η ( x ) for any x ∈ u ⋄ , and h t ηh † t ( E n ) = η ( E n ) ± tφ ( γ ). Itfollows Π − Π( η ) is contained in H λ (cid:5) λ . The Lemma 15 implies statement1). Statement 2) follows from Lemma 14. ✷ By the linear form λ = λ D,φ , we define a linear character of the subgroup U λ as follows ξ λ ( u ) = ε λ ( f ( u )) . Consider the induced character χ λ = Ind( ξ λ , U λ , U ) . Theorem 18 . If λ = λ D,φ , then χ λ ( u ) = | H λ (cid:5) λ || G ◦ (cid:5) λ | X µ ∈ G ◦ (cid:5) λ ε µ ( f ( u )) . Proof . Let ˙ ξ λ define the function on the group U equal to ξ λ on U λ and zerooutside of U λ . By definition χ λ ( u ) = 1 | U λ | X v ∈ U ˙ ξ λ ( vuv − ) . pplying Lemma 17 for all x ∈ u , we have X µ ∈ π − π ( λ ) ε µ ( x ) = ε λ ( x ) · X ν ∈ u ⊥ λ ε ν ( x ) = ( | U || U λ | · ε λ ( x ) , if x ∈ u λ ;0 , if x / ∈ u λ . Hence ˙ ξ λ ( u ) = | U λ || U | X µ ∈ π − π ( λ ) ε µ ( f ( u )) = | U λ || U | X µ ∈ H λ (cid:5) λ ε µ ( f ( u )) . Then χ λ ( u ) = 1 | U | · X µ ∈ H λ (cid:5) λ, v ∈ U ε µ ( vf ( u ) v − ) = | H λ (cid:5) λ || U | · | H λ | X h ∈ H λ , v ∈ U ε h (cid:5) λ ( vf ( u ) v − ) = | H λ (cid:5) λ || U | · | H λ | X h ∈ H λ , v ∈ U ε λ ( hvf ( u ) v † h † ) . Since G ◦ = U H λ , we get χ λ ( u ) = | H λ (cid:5) λ | · | U ∩ H λ || U | · | H λ | X g ∈ G ◦ ε λ ( gf ( u ) g † ) = | H λ (cid:5) λ || G ◦ | X g ∈ G ◦ ε λ ( gf ( u ) g † ) = | H λ (cid:5) λ || G ◦ (cid:5) λ | X µ ∈ G ◦ (cid:5) λ ε µ ( f ( u )) . ✷ Theorem 19 . Let U be the Sylow subgroup in orthogonal or symplecticgroup. The system of characters { χ λ } , and the partition of the group U into classes { K ( u ) } , where λ and u run through the sets of representativesof equivalence classes λ D ,φ ∈ u ∗ and u D ,φ ∈ U , give rise to a supercharactertheory of the group U . Proof . The proof follows from Remark 6, Theorems 13 and 18. ✷ References [1] P. Diaconis, I. M. Isaacs,
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